Evaluating Local Community Methods in Networks

We present a new benchmarking procedure that is unambiguous and specific to local community-finding methods, allowing one to compare the accuracy of various methods. We apply this to new and existing algorithms. A simple class of synthetic benchmark networks is also developed, capable of testing properties specific to these local methods.Comment: 8 pages, 9 figures, code included with sourc

Weakly nonlinear quantum transport: an exactly solvable model

We have studied the weakly non-linear quantum transport properties of a two-dimensional quantum wire which can be solved exactly. The non-linear transport coefficients have been calculated and interesting physical properties revealed. In particular we found that as the incoming electron energy approaches a resonant point given by energy $E=E_r$, where the transport is characterized by a complete reflection, the second order non-linear conductance changes its sign. This has interesting implications to the current-voltage characteristics. We have also investigated the establishment of the gauge invariance condition. We found that for systems with a finite scattering region, correction terms to the theoretical formalism are needed to preserve the gauge invariance. These corrections were derived analytically for this model.Comment: 15 pages, LaTeX, submitted to Phys. Rev.

Bound States and Threshold Resonances in Quantum Wires with Circular Bends

We study the solutions to the wave equation in a two-dimensional tube of unit width comprised of two straight regions connected by a region of constant curvature. We introduce a numerical method which permits high accuracy at high curvature. We determine the bound state energies as well as the transmission and reflection matrices, ${\cal T}$ and ${\cal R}$ and focus on the nature of the resonances which occur in the vicinity of channel thresholds. We explore the dependence of these solutions on the curvature of the tube and angle of the bend and discuss several limiting cases where our numerical results confirm analytic predictions.Comment: 24 pages, revtex file, one style file and 17 PostScript figures include

A unified approach for the solution of the Fokker-Planck equation

This paper explores the use of a discrete singular convolution algorithm as a unified approach for numerical integration of the Fokker-Planck equation. The unified features of the discrete singular convolution algorithm are discussed. It is demonstrated that different implementations of the present algorithm, such as global, local, Galerkin, collocation, and finite difference, can be deduced from a single starting point. Three benchmark stochastic systems, the repulsive Wong process, the Black-Scholes equation and a genuine nonlinear model, are employed to illustrate the robustness and to test accuracy of the present approach for the solution of the Fokker-Planck equation via a time-dependent method. An additional example, the incompressible Euler equation, is used to further validate the present approach for more difficult problems. Numerical results indicate that the present unified approach is robust and accurate for solving the Fokker-Planck equation.Comment: 19 page

Current conservation in two-dimensional AC-transport

The electric current conservation in a two-dimensional quantum wire under a time dependent field is investigated. Such a conservation is obtained as the global density of states contribution to the emittance is balanced by the contribution due to the internal charge response inside the sample. However when the global partial density of states is approximately calculated using scattering matrix only, correction terms are needed to obtain precise current conservation. We have derived these corrections analytically using a specific two-dimensional system. We found that when the incident energy $E$ is near the first subband, our result reduces to the one-dimensional result. As $E$ approaches to the $n$-th subband with $n>1$, the correction term diverges. This explains the systematic deviation to precise current conservation observed in a previous numerical calculation.Comment: 12 pages Latex, submitted to Phys. Rev.

Kink propagation in a two-dimensional curved Josephson junction

We consider the propagation of sine-Gordon kinks in a planar curved strip as a model of nonlinear wave propagation in curved wave guides. The homogeneous Neumann transverse boundary conditions, in the curvilinear coordinates, allow to assume a homogeneous kink solution. Using a simple collective variable approach based on the kink coordinate, we show that curved regions act as potential barriers for the wave and determine the threshold velocity for the kink to cross. The analysis is confirmed by numerical solution of the 2D sine-Gordon equation.Comment: 8 pages, 4 figures (2 in color

Localization of nonlinear excitations in curved waveguides

Motivated by the example of a curved waveguide embedded in a photonic crystal, we examine the effects of geometry in a ``quantum channel'' of parabolic form. We study the linear case and derive exact as well as approximate expressions for the eigenvalues and eigenfunctions of the linear problem. We then proceed to the nonlinear setting and its stationary states in a number of limiting cases that allow for analytical treatment. The results of our analysis are used as initial conditions in direct numerical simulations of the nonlinear problem and localized excitations are found to persist, as well as to have interesting relaxational dynamics. Analogies of the present problem in contexts related to atomic physics and particularly to Bose-Einstein condensation are discussed.Comment: 14 pages, 4 figure

Hall-like effect induced by spin-orbit interaction

The effect of spin-orbit interaction on electron transport properties of a cross-junction structure is studied. It is shown that it results in spin polarization of left and right outgoing electron waves. Consequently, incoming electron wave of a proper polarization induces voltage drop perpendicularly to the direct current flow between source and drain of the considered four-terminal cross-structure. The resulting Hall-like resistance is estimated to be of the order of 10^-3 - 10^-2 h/e^2 for technologically available structures. The effect becomes more pronounced in the vicinity of resonances where Hall-like resistance changes its sign as function of the Fermi energy.Comment: 4 pages (RevTeX), 4 figures, will appear in Phys. Rev. Let